On the Classical Prime Spectrum of Lattice Modules

Abstract

Let M be a lattice module over a C-lattice L. A proper element P of M is said to be classical prime if for a, b ∈ L and X ∈ M, abX ≤ P implies that aX ≤ P or bX ≤ P. The set of all classical prime elements of M , Spec cp (M) is called as classical prime spectrum. In this article, we introduce and study a topology on Spec cp (M), called as Zariski-like topology of M. We investigate this topological space from the point of view of spectral spaces. We show that if M has ascending chain condition on classical prime radical elements, then Spec cp (M) with the Zariski-like topology is a spectral space.